HAMBURGER BEITRÄGE ZUR MATHEMATIK

Transcription

1 HAMBURGER BEITRÄGE ZUR MATHEMATI Heft 561 An algorthmc hypergraph regularty lemma Brendan Nagle, Tampa Vojtěch Rödl, Atlanta Mathas Schacht, Hamburg Verson September 2015

2 AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT Abstract. Szemeréd s Regularty Lemma s a powerful tools n graph theory. It asserts that all large graphs admt bounded parttons of ther edge sets, most classes of whch consst of unformly dstrbuted edges. The orgnal proof of ths result was non-constructve and a constructve proof was later gven by Alon, Duke, Lefmann, Rödl and Yuster. Szemeréd s Regularty Lemma was extended to hypergraphs by varous authors. Frankl and Rödl gave one such extenson n the case of 3-unform hypergraphs, whch was later extended to k-unform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extenson, usng a dfferent concept of regularty than that of Frankl, Rödl and Skokan. In ths paper, we gve a constructve proof of the Regularty Lemma for hypergraphs. 1. Introducton Szemeréd s Regularty Lemma [21, 22] s an mportant tool n combnatorcs, wth applcatons rangng across combnatoral number theory, extremal graph theory, and theoretcal computer scence (see [9, 10] for surveys of applcatons. The Regularty Lemma hnges on the noton of ε-regularty: a bpartte graph H = (X Y, E s ε-regular f for every X X, X > ε X, and for every Y Y, Y > ε Y, we have d H (X, Y d H (X, Y < ε, where d H (X, Y = H[X, Y ] /( X Y s the densty of the bpartte graph H[X, Y ] nduced on the sets X and Y. Szemeréd s Regularty Lemma s then stated as follows. Theorem 1.1 (Szemeréd s Regularty Lemma [21, 22]. For all ε > 0 and ntegers t 0 1, there exst ntegers T 0 = T 0 (ε, t 0 and N 0 = N 0 (ε, t 0 so that every graph G on N > n 0 vertces admts a partton of ts vertex set V (G = V 1 V t, t 0 t T 0, satsfyng (1 V (G = V 1 V t s equtable: V 1 V t V 1 + 1; (2 V (G = V 1 V t s ε-regular: all but ε ( t 2 pars V, V j, 1 < j t, are ε-regular. A constructve proof of Theorem 1.1 was later gven by Alon, Duke, Lefmann, Rödl and Yuster. Ther result gves that the ε-regular partton V (G = V 1 V t n Theorem 1.1 can be constructed n tme O(M(n, where M(n = O(n s the tme needed to multply two n n matrces wth 0, 1-entres over the ntegers (see [23]. In [8], the runnng tme of O(M(n was mproved to O(n 2. Szemeréd s Regularty Lemma has been extended to k-unform hypergraphs, for k 2, by varous authors. Frankl and Rödl [3] gave one such extenson to the case of 3-unform hypergraphs, usng a concept they called (δ, r-regularty (see upcomng Defnton Ths regularty lemma was extended to k-unform hypergraphs, for arbtrary k 3, by Rödl and Skokan [17]. Gowers [4, 5] also establshed a regularty lemma for k-unform hypergraphs, but used a concept of regularty known as devaton (see upcomng Defnton 2.6. Whle the concepts of (δ, r-regularty and devaton are dfferent, the correspondng Regularty Lemmas have a smlar concluson. Roughly speakng, both lemmas guarantee that every (large k-unform hypergraph admts a bounded partton of ts edge set (where the edge-partton s defned n a farly techncal way, where most classes of the partton consst of regularly dstrbuted 1

3 2 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT edges. Moreover, both Regularty Lemmas admt a correspondng Countng Lemma (see upcomng Theorems 5.1 and 5.2, and see also [12]. The Countng Lemma allows one to estmate the number of fxed subhypergraphs of a gven somorphsm type wthn the regular partton a regularty lemma provdes. The combned applcaton of the Regularty and Countng Lemmas s known as the Regularty Method for hypergraphs (see [13, 15, 16, 19] for surveys of applcatons. The goal of ths paper s to establsh an algorthmc Hypergraph Regularty Lemma (see upcomng Theorem 3.7. Roughly speakng, we wll show that, for every (large k-unform hypergraph H (k, a regular partton of H (k can, n fact, be constructed n tme polynomal n V (H (k. Thus, combnng the work here together wth an approprate Countng Lemma provdes an Algorthmc Regularty Method for hypergraphs 1. To prove the algorthmc regularty lemma for hypergraphs, we wll proceed along the usual lnes. As n the proof of Szemeréd [21, 22] for graphs, we wll consder sequences of parttons P, 1, of a hypergraph H (k. For each P, 1, we consder the so-called ndex of P, denoted nd H (k(p, whch measures the mean-square densty of H (k on P. When the partton P of H (k s rregular, we refne P, n the usual way, to produce P +1. It s wellknown that nd H (k(p +1 wll be non-neglgbly larger than nd H (k(p, so that ths refnng process must termnate after constantly many teratons. Now, as n the proof of Alon et al. [1] for graphs, to make the the refnement P +1 of P constructve, one must be able to construct wtnesses of the rregularty of P. The novel element of our work does precsely ths and n Secton 2, we state the Wtness-Constructon Theorem (Theorem In Secton 3, we state the Algorthmc Regularty Lemma (Theorem 3.7, and n Secton 4, we show that Theorem 2.16 mples Theorem 3.7. The remander of the paper s devoted to provng Theorem For ths proof, we wll need several techncal lemmas. Among these are Gowers Countng Lemma (see Theorems 5.1 and 5.2, whch we present n Secton 5. As well, we wll need an Extenson Lemma (Theorem 5.4, whch s a dervatve of the Countng Lemma, whch we also present n Secton 5. Fnally, we need an addtonal lemma, whch we call the Negatve-Extenson Lemma (Theorem 6.2, whch we state and prove n Secton 6. Usng these tools, we prove Theorem 2.16 n Secton 7. At the end of the paper, we nclude an Appendx for the proofs of a few facts we need along the way. 2. Devaton and the Wtness-Constructon Theorem In ths secton, we defne the concept of devaton (DEV (cf. Defnton 2.6, and we present some condtons whch are suffcent for mplyng the property of devaton. We also consder the concept of r-dscrepancy (r-disc (cf. Defnton 2.12, and present a so-called Wtness- Constructon theorem (cf. Theorem For these purposes, we need some supportng concepts Background concepts: cylnders, complexes and densty. We begn wth some basc concepts. For a set X and an nteger j X, let ( X j denote the set of all (unordered j-tuples from X. When X = [l] = {1,..., l}, we sometmes wrte [l] j = ( [l] j. Gven parwse dsjont sets V 1,..., V l, denote by (j (V 1,..., V l the complete l-partte, j-unform hypergraph 1 An algorthmc regularty method for 3-unform hypergraphs was establshed by Haxell, Nagle, and Rödl [6, 7] (see also [11].

4 AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA 3 wth l-partton V 1 V l, whch conssts of all j-tuples from V 1 V l meetng each V a, 1 a j, at most once. We now defne the concept of a cylnder. Defnton 2.1 (cylnder. For ntegers l j 1, an (l, j-cylnder H (j wth vertex l- partton V (H (j = V 1 V l s any subset of (j (V 1,..., V l. When V 1 = = V l = m, we say H (j s an (m, l, j-cylnder. In the context of Defnton 2.1, fx j l and Λ [l]. We denote by H (j [Λ ] = H (j[ ] λ Λ V λ the sub-hypergraph of the (l, j-cylnder H (j nduced on λ Λ V λ. In ths settng, H (j [Λ ] s an (, j-cylnder. We now prepare to defne the concept of a complex. For an nteger j, let (H (j denote the famly of all -element subsets of V (H (j whch span complete sub-hypergraphs n H (j. Gven an (l, j 1-cylnder H (j 1 and an (l, j-cylnder H (j, we say H (j 1 underles H (j f H (j j (H (j 1. Defnton 2.2 (complex. For ntegers 1 k l, an (l, k-complex H = {H (j } k j=1 s a collecton of (l, j-cylnders, 1 j k, so that (1 H (1 = V 1 V l s an (l, 1-cylnder,.e., s an l-partton; (2 for each 2 j k, we have that H (j 1 underles H (j,.e., H (j j (H (j 1. In some cases, we use the notaton H (k to denote an (l, k-complex {H (j } k j=1. We now defne concept of densty. Defnton 2.3 (densty. For ntegers 2 j l, let H (j be an (l, j-cylnder and let H (j 1 be an (l, j 1-cylnder. If j (H (j 1, we defne the densty of H (j w.r.t. H (j 1 as H d(h (j H (j 1 (j j (H (j 1 = j (H (j 1. If j (H (j 1 =, we defne d(h (j H (j 1 = Devaton, and suffcent condtons thereof. In ths subsecton, we defne the concept of devaton (DEV, and present some condtons whch are suffcent for mplyng the property of devaton. To that end, we need some supportng concepts. Defnton 2.4 ((l, j-octohedron. Let ntegers 1 j l be gven. The (l, j-octohedron O (j = O (j l s the complete l-partte j-unform hypergraph (j (U 1,..., U l, where U 1 = = U l = 2,.e., t s the complete (2, l, j-cylnder. For an (l, j-cylnder H (j, we are nterested n labeled partte-embedded copes of O (j n H (j. Defnton 2.5 (labeled partte-embeddng. Let H (j be an (l, j-cylnder, wth l-partton V (H (j = V 1 V l, and let O (j = (j (U 1,..., U l be the (l, j-octohedron. A labeled, partte-embeddng of O (j n H (j s an edge-preservng njecton ψ : U 1 U l V 1 V l so that ψ(u V for each 1 l. We wrte EMB(O (j, H (j to denote the famly of all labeled partte-embeddngs ψ of O (j n H (j. We now defne the concept of devaton.

5 4 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT Defnton 2.6 (devaton (DEV. Let H (j be a (j, j-cylnder wth underlyng (j, j 1- cylnder H (j 1. Let H (j and H (j 1 have common vertex j-partton V (H (j = V (H (j 1 = V 1 V j, and let d = d(h (j H (j 1. For δ > 0, we say that (H (j, H (j 1 has (d, δ- devaton, wrtten DEV(d, δ, f { ω(j : J (j ( {v 1, v 1},..., {v j, v j} } δ EMB(O (j 1, H (j 1, v 1,v 1 V 1 v j,v j V j ( where for every v 1, v 1 V 1,..., v j, v j V j, and for each J (j {v 1, v 1 },..., {v j, v j, } 1 d f J H (j, ω(j = d 0 f J j (H (j 1 \ H (j, f J j (H (j 1. It s easy to extend Defnton 2.6 from (j, j-cylnders to (l, k-complexes. Defnton 2.7. Let δ = (δ 2,..., δ k and d = (d Λj : [l] j, 2 j k be sequences of postve reals, and let (l, k-complex H = {H (j } k j=1 be gven. We say the complex H has DEV(d, δ f, for each 2 j h and [l] j, (H (j [ ], H (j 1 [ ] has DEV(d Λj, δ j. For future reference, we present some easy suffcent condtons for the property of devaton (cf. Defnton 2.6. For that, we need the followng generalzaton of Defnton 2.5. Defnton 2.8 (labeled partte-embeddng. Let H (j and H (j 1 be gven as n Defnton 2.6, and let S (j O (j = (j (U 1,..., U j be an arbtrary (2, j, j-cylnder. We call an njecton ψ : U 1 U j V 1 V j a labeled partte-embeddng of S (j n (H (j, H (j 1 f t satsfes the followng condtons: (1 ψ s a labeled partte-embeddng of O (j 1 = (j 1 (U 1,..., U j n H (j 1 ; (2 for each J O (j = (j (U 1,..., U j, we have J S (j = ψ(j H (j. We call ψ labeled, partte-nduced embeddng of S (j n (H (j, H (j 1 f t satsfes (1 and (2 above, together wth (2 for each J O (j = (j (U 1,..., U l, we have J S (j ψ(j H (j. We wrte EMB(S (j, (H (j, H (j 1 to denote the famly of all labeled partte-embeddngs ψ of S (j n (H (j, H (j 1. We wrte EMB nd (S (j, (H (j, H (j 1 to denote the famly of all labeled, partte-nduced embeddngs ψ of S (j n (H (j, H (j 1. We now consder the followng two propertes. Defnton 2.9 (COUNT emb, COUNT nd. Let H (j and H (j 1 be gven as n Defnton 2.8, where d = d(h (j H (j 1. For δ > 0, we say that (H (j, H (j 1 has COUNT emb (d, δ f the followng condton holds: for every (2, j, j-cylnder S (j O (j = (j (U 1,..., U j, we have EMB(S (j, (H (j, H (j 1 = (1 ± δd S (j EMB(O (j 1, H (j 1. (1 (Note that when S (j =, t always holds that EMB(, (H (j, H (j 1 = (1 ± δd 0 EMB(O (j 1, H (j 1, (2

6 AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA 5 snce every labeled partte-embeddng ψ of n H (j s, equvalently, a labeled partte-embeddng of O (j 1 n H (j 1. We say that (H (j, H (j 1 has COUNT nd (d, δ f the followng condton holds: for every (2, j, j-cylnder S (j O (j = (j (U 1,..., U j, EMB nd (S (j, (H (j, H (j 1 = (1 ± δd S(j (1 d 2j S (j EMB(O (j 1, H (j 1. The followng fact wll be useful later n ths paper. The proof s easy, and we gve t n the Appendx. Fact Suppose H (j and H (j 1 are gven as n Defnton 2.9, where d = d(h (j H (j 1 > 0, and let δ > 0 be gven. Suppose, moreover, that EMB(O (j 1, H (j 1 = Ω(n 2j, where V = Θ(n for all [j]. (1 (H (j, H (j 1 has COUNT emb (d, δ f, and only f, (H (j, H (j 1 has COUNT nd (d, δ; (2 If (H (j, H (j 1 has COUNT emb (d, δ, then (H (j, H (j 1 has DEV(d, δ r-dscrepancy, and the Wtness-Constructon Theorem. In ths subsecton, we defne the concept of r-dscrepancy (r-disc, and present the Wtness Constructon Theorem (cf. Theorem We begn wth the followng extenson of the concept of densty (cf. Defnton 2.3. Defnton 2.11 (r-densty. Let H (j and H (j 1 be gven as n Defnton 2.3, and let nteger r 1 be gven. Let Q (j 1 1,..., Q (j 1 r H (j 1 satsfy [r] j(q (j 1. We defne the r-densty of H (j w.r.t. Q (j 1 1,..., Q r (j 1 as d(h (j Q (j 1 1,..., Q (j 1 r = We now defne the concept of r-dscrepancy. H (j [r] j(q (j 1 [r] j(q (j 1. Defnton 2.12 (r-dscrepancy (r-disc. Let H (j and H (j 1 be gven as n Defnton 2.3, where d = d(h (j H (j 1. For δ > 0 and an nteger r 1, we say that (H (j, H (j 1 has (d, δ, r-dscrepancy, wrtten DISC(d, δ, r, f for any collecton Q (j 1 1,..., Q (j 1 r H (j 1, [r] j (Q (j 1 > δ j (H (j 1 = d(h (j Q (j 1 1,..., Q (j 1 r d < δ. (3 For brevty, we sometmes refer to (d, δ, r-dscrepancy as r-dscrepancy, and sometmes wrte DISC(d, δ, r as r-disc. We proceed wth the followng remark. Remark Note that 1-dscrepancy s usually referred to as dscrepancy, and 1-DISC s usually denoted by DISC (cf. [11]. We wll also need the followng concept, related to Defnton Defnton 2.14 (r-wtness. Let H (j and H (j 1 be gven as n Defnton 2.12, where d = d(h (j H (j 1. Suppose that (H (j, H (j 1 does not have DISC(d, δ, r, for some δ > 0 and nteger r 1. We call any collecton Q (j 1 1,..., Q (j 1 r H (j 1 for whch [r] j (Q (j 1 > δ j (H (j 1 but an r-wtness of DISC(d, δ, r. d(h (j Q (j 1 1,..., Q (j 1 r d δ.

7 6 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT We fnally present the Wtness-Constructon Theorem, whch concerns a (k, k-complex H satsfyng the followng setup. Setup Let H = H (k = {H (j } k j=1 be a (k, k-complex, where H(1 = V 1 V k has n V n + 1 for all [k]. Let d k = ( d Λj : [k] j, 2 j k satsfy that, for each 2 j k and for each [k] j, d Λj = d(h (j [ ] H (j 1 [ ]. Note, n partcular, that d [k] = d(h (k H (. We call d k the densty sequence for H (k. Wrte H ( = {H (j } j=1 and d = ( d Λj : [k] j, 2 j k 1, so that d s the densty sequence for H (. The Wtness-Constructon Theorem s now gven as follows. Theorem 2.16 (Wtness-Constructon Theorem. Let nteger k 2 be fxed. For all d k, δ k > 0, there exsts δ k > 0 so that for all d > 0, there exsts δ > 0 so that,..., for all d 2 > 0, there exst δ 2 > 0, postve nteger r 0, and postve nteger n 0 so that the followng holds. Set δ = (δ 2,..., δ. Let H = H (k be a (k, k-complex wth densty sequence d k, as gven as n Setup 2.15, where n n 0. Suppose d k satsfes that, for each 2 j k and for each [k] j, d Λj d j. Assume that (1 H ( has DEV(d, δ, but that (2 (H (k, H ( does not have DEV(d [k], δ k. Then, there exsts an algorthm whch constructs, n tme O(n 3k, an r-wtness Q ( 1,..., Q ( r H ( of DISC(d [k], δ k, r, for some r r Algorthmc Hypergraph Regularty Lemma In ths secton, we state an Algorthmc Hypergraph Regularty Lemma (see Theorem 3.7, below for the property of devaton. To state ths lemma, we stll need some more concepts Famles of parttons. Theorem 3.7 provdes a well-structured famly of parttons P = {P (1,..., P ( } of vertces, pars,..., and (-tuples of a gven vertex set. We wll defne the propertes of P n upcomng Defntons 3.1 and 3.2, but we frst need to establsh some notaton and concepts. We frst dscuss the structure of these parttons nductvely, followng the approach of [12]. Let k be a fxed nteger and V be a set of vertces. Let P (1 = {V 1,..., V P (1 } be a partton of V. For every 1 j P (1, let Cross j (P (1 = (j (V 1,..., V P (1 be the famly of all crossng j-tuples J,.e., the set of j-tuples whch satsfy J V 1 for every 1 P (1. Suppose that parttons P ( of Cross (P (1 have been defned for all 1 j 1. Then for every I Cross j 1 (P (1, there exsts a unque class P (j 1 = P (j 1 (I P (j 1 so that I P (j 1. For every J Cross j (P (1, we defne the polyad of J by ˆP (j 1 (J = { P (j 1 (I: I [J] j 1}. Defne the famly of all polyads P ˆ (j 1 = { ˆP(j 1 (J: J Cross j (P (1 }, whch we vew as a set (as opposed to a multset, snce ˆP (j 1 (J = ˆP (j 1 (J

8 AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA 7 may hold for J J. To smplfy notaton, we often wrte the elements of P ˆ (j 1 as ˆP (j 1 P ˆ(j 1 (droppng the argument J. Observe that { j ( ˆP (j 1 : ˆP(j 1 P ˆ(j 1 } s a partton of Cross j (P (1. The structural requrement on the partton P (j of Cross j (P (1 s P (j { j ( ˆP (j 1 : ˆP(j 1 ˆ P (j 1 }, (4 where denotes the refnement relaton of set parttons. Note that (4 nductvely mples that P(J = { ˆP( (J } j 1 =1, where ˆP ( (J = { P ( (I: I [J] }, (5 s a (j, j 1-complex (snce each ˆP ( (J s a (j, -cylnder. We may now gve Defntons 3.1 and 3.2. Defnton 3.1 (a-famly of parttons. Let V be a set of vertces, and let k 2 be a fxed nteger. Let a = (a 1,..., a be a sequence of postve ntegers. We say P = P(k 1, a = {P (1,..., P ( } s an a-famly of parttons on V, f t satsfes the followng: (a P (1 s a partton of V nto a 1 classes, (b P (j s a partton of Cross j (P (1 refnng { j ( ˆP (j 1 : ˆP(j 1 P ˆ(j 1 } where, for every ˆP (j 1 P ˆ(j 1, {P (j P (j : P (j j ( ˆP (j 1 } = a j. Moreover, we say P = P(k 1, a s t-bounded, f max{a 1,..., a } t Propertes of famles of parttons. In ths subsecton, we descrbe some propertes we would lke an a-famly of parttons P = P(k 1, a to have. Defnton 3.2 ((η, δ, D, a-famly. Let V be a set vertces, let η > 0 be fxed, and let k 2 be a fxed nteger. Let δ = (δ 2,..., δ and D = (D 2,..., D be sequences of postves, and let a = (a 1,..., a be a sequence of postve ntegers. We say an a-famly of parttons P = P(k 1, a on V s an (η, δ, D, a-famly f t satsfes the followng condtons: (a P (1 = {V : [a 1 ]} s an equtable vertex partton,.e., V /a 1 V V /a 1 for [a 1 ]; (b [V ] k \ Cross k (P (1 η V k ; (c all but η V k many k-tuples Cross k (P (1 satsfy that for each 2 j k 1, and for each J ( j, the par (P (j (J, ˆP (j 1 (J has DEV(d J, δ j, where d J = d(p (j (J ˆP (j 1 (J D j. Note that n an (η, δ, D, a-famly of parttons P on V, propertes (b and (c above mply that all but 2η V k many k-tuples [V ] k belong to Cross k (P (1 and satsfy that, for each 2 j k 1, and for each J ( j, the par (P (j (J, ˆP (j 1 (J has DEV(d J, δ j, where d J = d(p (j (J ˆP (j 1 (J D j. For future reference, we also defne the followng concept, related to property (c n Defnton 3.2. Defnton 3.3 ((δ, D-typcal polyad. Suppose P = P(k 1, a s an (η, δ, D, a- famly of parttons on a vertex set V, where δ = (δ 2,..., δ and D = (D 2,..., D. We say a polyad ˆP ( P ˆ( s (δ, D-typcal f (a k ( ˆP (, and fxng any k ( ˆP (, f

9 8 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT (b the correspondng (k, k 1-complex P( (cf. (5 satsfes that, for each 2 j k 1, and for each J ( j, the par (P (j (J, ˆP (j 1 (J has DEV(d J, δ j, where d J = d(p (j (J ˆP (j 1 (J D j. Remark 3.4. Note that property (c of Defnton 3.2 can be re-wrtten as { k ( ˆP ( : ˆP( P ˆ } ( s not (δ, D-typcal η V k. Note that n an (η, δ, D, a-famly P = {P (1,..., P ( } (cf. Defnton 3.2, the vertces, pars,..., and (k 1-tuples of V are under regular control. The followng defnton descrbes how the famly P wll control the edges of a hypergraph H (k, where V = V (H (k. Defnton 3.5 ((H (k, P has DEV(δ k. Let δ k > 0 be gven. For a k-graph H (k and an a-famly of parttons P = P(k 1, a on V = V (H (k, we say (H (k, P has DEV(δ k f { k ( ˆP ( : ˆP( P ˆ( satsfes that (H (k, ˆP ( does not have DEV ( d(h (k ˆP(, δ k } δ k V k. Before we state the algorthmc hypergraph regularty lemma, we say a word about some notaton we use n t. Remark 3.6. Let D = (D 2,..., D (0, 1] be a sequence, and for each 2 k 1, let δ : (0, 1] k (0, 1 be a functon (of k many (0, 1] varables, where we wrte δ = (δ 2,..., δ. We shall use the notaton δ(d = (δ (D,..., D : 2 k 1 to denote the sequence of functon values whose th coordnate, 2, s δ (D,..., D. We consder ths concept snce, n most applcatons of Theorem 3.7, one needs the value δ to be suffcently small not only w.r.t. D, but also D +1,..., D. We now state the algorthmc hypergraph regularty lemma. Theorem 3.7 (Algorthmc Hypergraph Regularty Lemma. Let k 2 be a fxed nteger, and let η, δ k > 0 be fxed postves. For each 2 k 1, let δ : (0, 1] k (0, 1 be a functon, and set δ = (δ 2,..., δ. Then, there exst t, n 0 N so that the followng holds. For every k-unform hypergraph H (k wth V (H (k = n n 0, one may construct, n tme O(n 3k, a famly of parttons P = P(k 1, a P of V (H (k wth the followng propertes: ( P s a t-bounded (η, δ(d, D, a P -famly on V (H (k (cf. Remark 3.6; ( (H (k, P has DEV(δ k. We proceed wth the followng remark. Remark 3.8. Smlarly as n Szemeréd [21, 22] for graphs, t s well-known that one can prove a hypergraph regularty lemma whch regularzes not one, but multple hypergraphs H (k 1,..., H(k s (on a common vertex set V smultaneously. More precsely, n the context of Theorem 3.7, the t-bounded (η, δ(d, D, a P -famly above wll satsfy that, for each 1 s, the par (H (k, P has DEV(δ k, where t = t(s, k, η, δ k, δ and V n 0 = n 0 (s, k, η, δ k, δ. We shall prove Theorem 3.7 by nducton on k 2. To avod formalsm, we shall be provng the case s = 1, but our nducton hypothess wll assume the general case.

10 AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA 9 4. Proof of Theorem 3.7 The proof of Theorem 3.7 s by nducton on k 2. The nducton begns wth k = 2 as a known base case. Indeed, Alon et al. [1] proved an algorthmc verson of the Szemeréd Regularty Lemma, whch s Theorem 3.7 (k = 2 wth DEV replaced by DISC. Gowers [4, 5] proved that DEV and DISC are equvalent propertes when k = 2, and so the base case of Theorem 3.7 holds. We assume Theorem 3.7 holds through k 1 2, and prove t for k 3. To that end, we need a few supportng consderatons Supportng materal. Suppose H (k s a k-unform hypergraph wth vertex set V = V (H (k, where V = n. Let P = P(k 1, a be an a-famly of parttons on V. We defne the ndex of P w.r.t. H (k as Clearly, nd H (k(p = 1 n k {d 2 (H (k ˆP ( k ( ˆP ( : ˆP( ˆ P (}. 0 nd H (k(p 1. (6 The proof of Theorem 3.7 s smlar to that of Szemeréd [21, 22], where we wll use the followng so-called Index-pumpng Lemma (Lemma 4.1 below. To ntroduce ths lemma, let H (k be a k-unform hypergraph wth vertex set V = V (H (k, where V = n. Snce ths proof s by nducton on k, suppose we already have a regular partton P = P(k 1, a of V up through k 1. More precsely, let P = P(k 1, a be an arbtrary t-bounded, (η, δ(d, D, a-famly on V. We now test how H (k behaves on P. In partcular, we test whether (H (k, P has DEV(δ k, whch we may do n tme O(n 2k. Indeed, for each polyad ˆP ( P ˆ(, we test (by usng Defnton 2.6 whether or not (H (k, ˆP ( has DEV(d ˆP(, δ k, where d = d(h (k ˆP (. ˆP( We arrve at two cases. Case 1. Suppose we fnd that most polyads ˆP ( P ˆ( satsfy that (H (k, ˆP ( has DEV(d ˆP(, δ k. Then we stop, and P s the partton we seek n Theorem 3.7. Case 2. Suppose we fnd many polyads ˆP ( P ˆ( for whch the par (H (k, ˆP ( fals to have DEV(d ˆP(, δ k. Then, for each such ˆP ( P ˆ(, Theorem 2.16 bulds (n tme O(n 3k an r ˆP(-wtness Q ( = {Q( ˆP ( 1,..., Q ( r } ˆP( of DISC(d ˆP(, δ k, r ˆP(, where δ k = δ k (δ k > 0 depends on δ k, and where r ˆP( r(d, where r(d depends on D. Now, Lemma 4.1 (below constructs, n tme O(n, a new partton P from P and the wtnesses Q ( ˆP (, over those polyads ˆP ( P ˆ( falng to have DEV(d ˆP(, δ k, where nd H (k( ˆ P nd H (k(p + δ 4 k 2. We now state the Index-pumpng Lemma precsely.

11 10 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT Lemma 4.1 (Index-pumpng Lemma. Fx an nteger k 2, and let ν, δ k > 0 be fxed. For each 2 k 1, let δ : (0, 1] k (0, 1 be a functon, where we set δ = (δ 2,..., δ. Let r : (0, 1] k 2 N be an arbtrary functon. Let D old = (D2 old,..., Dold (0, 1]k 2 and a old = (a old 1,..., aold N be fxed. Then, there exst D new = (D2 new,..., D new a new = (a new 1,..., a new N, and n 0 N so that the followng holds. (0, 1]k 2, Suppose H (k s a k-unform hypergraph wth vertex set V = V (H (k, where V = n n 0. Suppose P old = P old (k 1, a s a t old -bounded (ν, δ(d old, D old, a old -famly on V, where t old = max{a old 1,..., aold } and where δ(d old = ( δ (D old,..., D old. Suppose that ˆ P ( ˆ =2 P ( s a gven collecton of polyads satsfyng the followng propertes: (1 ˆP ( ˆ (2 Then, P (, one s gven an r ˆP(-wtness Q ( ˆP ( of DISC(d ˆP(, δ k, r ˆP(, where r ˆP( r(d old = r(d old 2,..., Dold ; { k ( ˆP ( : ˆP( ˆ P ( } δ k n k. (a there exsts a t new -bounded (ν, δ(d new, D new, a new -famly P new = P new (, a new on V for whch nd H (k(p new nd H (k(p old + δ 4 k 2, where t new = max{a new 1,..., a new } and where δ(d new = ( δ (D new,..., D new =2. (b Moreover, there exsts an algorthm whch, n tme O(n, constructs the partton P new above from P old and the gven collecton of wtnesses { Q ( ˆP ( : ˆP ( ˆ P ( }. Lemma 4.1 s essentally gven as Lemma 8.3 of [17] and Lemma 6.3 of [5]. The proof of Lemma 4.1 s gven n [5, 17], but wth no focus to beng algorthmc. We shall not gve a formal proof of Lemma 4.1, but we wll sketch a proof to ndcate how ts algorthmc part s obtaned. Indeed, the approach n [17] s smlar to Szemeréd s [21, 22]. Consder the Venn Dagram of the ntersectons of the r ˆP(-wtnesses Q ( ˆP (, over ˆP ( P ˆ (. By Statement (1 n the hypothess of Lemma 4.1, these wtnesses are gven to us. (In [17], these wtnesses are assumed to exst, but here, we wll buld them wth Theorem Ths Venn dagram has at most ( ˆP r(d 2 old regons (ths number s ndependent of n, where each regon s a (k 1, k 1-cylnder. Ths Venn Dagram defnes a refnement P old of P old, so that P old s tself a partton. The ndex of P old wll be larger than that of P old on account of the fact that, n Statement (2, we assumed many k-tuples were lost to polyads ˆP ( P ˆ (. The (k 1, k 1-cylnders of P old may not have DEV(δ k, so we apply Theorem 3.7 to each (where we assume, by nducton on k, that Theorem 3.7 s algorthmc for k 1 (cf. Remark 3.8. Ths process produces the partton P new, where t s well-known that, as a refnement of P old, we have nd H (k(p new nd H (k(p old. For the formal detals of ths outlne, see [5, 17].

12 AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA Proof of Theorem 3.7. The proof of Theorem 3.7 was casually revealed when we ntroduced the Index-pumpng Lemma. Here, we proceed wth the formal detals. Let η, δ k > 0 be gven. For each 2 k 1, let δ : (0, 1] k (0, 1 be a functon, and set δ = (δ 2,..., δ. We begn our argument by defnng some auxlary parameters Auxlary parameters for Theorem 3.7. In all that follows, set Let d k = δ k = ν = 1 3 mn{δ k, η} and t 0 = 2/ν. (7 δ k = δ k,thm.2.16 (d k, δ k (8 be the constant guaranteed by the Wtness-Constructon Theorem (Theorem More generally, recall that Theorem 2.16 has the followng quantfcaton: k, d k, δ k, δ k : d, δ :... d 2, δ 2, r 0, n 0 :... Ths means that for each 2 k 1, the constant δ (whch s guaranteed to exst by Theorem 2.16 depends on d j, for all j k 1 (whch were gven earler. In other words, Theorem 2.16 guarantees the exstence of the followng functon δ,thm.2.16 (d k, x,..., x : {d k } (0, 1] k 1 (0, 1 (9 where x = d,..., x = d (0, 1] are varables. Smlarly, wth varables x = d,..., x 2 = d 2 (0, 1], let r 0 (d k, x,..., x 2 : {d k } (0, 1] k 2 N (10 be the functon guaranteed by the Theorem We shall assume, w.l.o.g., that for each 2 k 1 and for every x,..., x (0, 1], we have δ (x,..., x δ,thm.2.16 (d k, x,..., x. (11 Indeed, for otherwse, we would replace the gven functon δ wth the functon δ,thm.2.16 and produce a partton P whch s more regular than was sought. In what follows, we set δ = (δ 2,..., δ k, and we emphasze that, n what follows, k, ν, δ k, δ, and r are fxed (as a result of (7 (11. (12 It remans to defne the promsed nteger t. Smlarly as n the proof of Szemeréd [21, 22], ths nteger wll be determned by an teratve procedure usng the Index-pumpng Lemma (Lemma 4.1. To that end, recall that Lemma 4.1 has the followng quantfcaton: k, ν, δ k, δ, r, D old, a old, D new, a new, n 0 :... We apply Lemma 4.1 wth the fxed choces k, ν, δ k, δ, and r from (12 so that Lemma 4.1 defnes functons D new (D old, a old = D new (ν, δ k, δ, r = r 0, D old, a old (0, 1] k 2, and a new (D old, a old = a new (ν, δ k, δ, r = r 0, D old, a old N, (13 where D old (0, 1] k 2 and a old N are sequences of varables. (Henceforth, we make the abbrevatons D = D new and a = a new. Now, we successvely defne sequences D ( (0, 1] k 2 and a ( N, as follows. Wth t 0 gven n (7, set D (1 = (d 2 = 1,..., d = 1 and a (1 = (a (1 1 = t 0, a (1 2 = 1,..., a (1 = 1. (14

13 12 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT For 2, set (cf. (13 D ( = D(D ( 1, a ( 1 = (d ( 2,..., d(, a ( = a(d ( 1, a ( 1 = (recall the functons gven n (13. Set (cf. (8 t = ( a ( 1,..., a(, and max t, where stop = 1 stop t = max Ths concludes the descrpton of parameters we need to prove Theorem 3.7. { } a ( 1,..., a( (15 2. ( The argument (algorthm for Theorem 3.7. Let H (k be a k-unform hypergraph wth vertex set V = V (H (k, where we assume n = V s suffcently large. Our goal s to construct, n tme O(n 3k, a famly of parttons P = P(k 1, a P of V whch s t-bounded (cf. (16, whch s an (η, δ(d, D, a-famly, where (H (k, P has DEV(δ k, and where the sequences D and a wll be gven by D ( and a ( (cf. (14 and (15, resp., for some 1 stop (cf. (16. To begn, let V = V 1 V t0 (cf. (7 be a vertex partton satsfyng n/t 0 V n/t 0, for each 1 t 0. Let P 1 = {P (1 1,..., P( 1 } be an ntal famly of parttons, where for each 2 j, the partton P (j 1 conssts of the ( t 0j many (j, j-cylnders (j (V 1,..., V j, where 1 1 < < j t 0. Then, P 1 s a t 0 -bounded (ν, δ(d (1, D (1, a (1 -famly of parttons (cf. (14. Indeed, all but ( n/t0 t 0 n k 2 < nk 2 t 0 (7 < νn k many k-tuples ( V k belong to Crossk (P (1 1, and every Cross k(p (1 1 satsfes that, for every 2 j k 1, and for every J ( j, the par (P (j (J, ˆP (j 1 (J has DEV(1, 0 (cf. Condtons (a (c of Defnton 3.2. For an nteger 1 < stop (cf. (16, assume P 1,..., P are constructed famles of parttons of V, where P = P (k 1, a s a t -bounded (ν, δ(d (, D (, a ( -famly, (17 for D (, a ( and t gven n (14 (15. We proceed wth the followng Steps 1 4. Step 1. Identfy, n tme O(n 2k, the sets { P ˆ (, DEV = ˆP( P ˆ( : (H (k, ˆP ( does not have DEV(d(H (k ˆP (, δ k { P ˆ (,typ = ˆP( P ˆ( : ˆP( s (δ(d (, D ( -typcal (cf. Defnton 3.5 ˆ P (,atyp = { ˆP( Identfy, n tme O(n k, the sets (cf. (7 ˆ P (,dense = { ˆP( ˆ P ( : d(h (k ˆP ( d k } δ 4 k ˆ P ( : ˆP( s not (δ(d (, D ( -typcal, ˆ P (,sparse = ˆ P ( \ ˆ P (,dense. }, }, }.

14 Identfy, n tme O(1, the set (The last dentfcaton uses that ˆ AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA 13 P ˆ (, = P ˆ(, DEV ˆ P ( = O(1. P (,typ Step 2. Compute 2 the sum S = { k ( ˆP ( : ˆP ( ˆ P (,dense. (18 ˆ P (, If S δ k n k (cf. (7, we proceed to Step 3. If S < δ k n k, then we stop, and the promsed partton s P = P. Indeed, snce ˆ P (, DEV ˆ P (, we have (cf. Remark 3.4 { k ( ˆP ( : ˆP ( P ˆ } (, DEV S + { k ( ˆP ( : ˆP ( ˆ P (,atyp ˆ P (,atyp ˆ P (,sparse, }. } + { k ( ˆP ( : ˆP ( P ˆ } (,sparse < δ k n k + ηn k + d k n k (7 < δ k n k, so that P = P has property DEV(δ k (cf. Defnton 3.3. Moreover, snce P s a t -bounded (ν, δ(d (, D (, a ( -famly, wth ν < η (cf. (7, then t s also an (η, δ(d (, D (, a ( - famly (cf. (7, as desred. Step 3. If S δ k n k, then we wll apply Theorem 2.16 to each ˆP ( P ˆ(,. We frst verfy that the hypothess of Theorem 2.16 wll be satsfed. To that end, fx ˆP ( P ˆ(,, and let P be the correspondng (k, k 1-complex (cf. (5. In the context of Theorem 2.16, P plays the role of H (, and (H (k k ( ˆP ( P plays the role of H (k. Snce ˆP ( P ˆ( (18, P ˆ(,typ, we have that P s (δ(d (, D ( -typcal, or n other words (cf. Defnton 3.3, P has DEV(d ˆP(, δ(d ( for some densty sequence d whch s coordnate-wse at least D (. ˆP( Snce ˆP ( P ˆ( (18, ˆ P (, DEV ˆ P (,dense, we have that (H (k, ˆP ( does not have DEV(d(H (k ˆP (, δ k, where d(h (k ˆP ( d k. Moreover, we have chosen the constants d k, δ k and δ k (cf. (7 and (8 and the functons δ(d ( and r 0 (D ( = r 0 (d k, d (,..., d( 2 (cf. (9 (11 approprately for an applcaton of Theorem Thus, the hypothess of Theorem 2.16 s satsfed, and so Theorem 2.16 constructs, n tme O(n 3k, an r ˆP(-wtness Q ( ˆP (, gven by Q ( 1, ˆP (,..., Q( r ˆP ˆP (, (19 ˆP(, ( 2 Snce S = O(n k has O(log n many dgts, Step 2 s done n tme O(log n.

15 14 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT of DISC(d(H (k, ˆP (, δ k, r ˆP(, where r ˆP( r 0 (D (. Repeat the applcaton of Theorem 2.16 over all ˆP ( P ˆ(,. Step 4. If S δ k n k, then we wll apply Lemma 4.1 to the famly of parttons P and the collecton of wtnesses Q ( ˆP (, over all ˆP ( P ˆ(,#. We frst verfy that the hypothess of Lemma 4.1 wll be satsfed. Indeed, by our nducton hypothess n (17, P s a constructed t - bounded (ν, δ(d (, D (, a ( famly of parttons. Assumpton (1 of Lemma 4.1 s satsfed because the set P ˆ (, was constructed n Step 1 (cf. (18, and for each ˆP ( P ˆ(,, a correspondng r ˆP(-wtness Q ( was constructed n Step 3 (cf. (19. Assumpton (2 of ˆP ( Lemma 4.1 s satsfed because we assume S δ k n k, and so S = { k ( ˆP ( : ˆP ( P ˆ } ( δ k n k (8 δ k n k. Thus, Lemma 4.1 constructs, n tme O(n, a t +1 -bounded (ν, δ(d (+1, D (+1, a (+1 famly of parttons P +1, where t +1, D (+1, and a (+1 are gven n (15, for whch nd H (k(p +1 nd H (k(p + δ 4 k 2. Return to Step 1 wth the newly constructed famly P +1. From (6, we may repeat Steps 1 4 above at most stop = 2/ δ k 4 tmes (cf. (16, whch proves Theorem Countng and Extenson Lemmas In ths secton, we present Countng and Extenson Lemmas for regular complexes. All results n ths secton can be derved, n a standard way, from the followng Countng Lemma for clques due to Gowers [4, 5], Theorem 5.1 (Clque Countng Lemma, Gowers. Let ntegers l k 2 be fxed. For all µ, d k > 0, there exsts δ k > 0 so that for all d > 0, there exsts δ > 0 so that,..., for all d 2 > 0, there exsts δ 2 > 0 and postve nteger n 0 so that the followng holds. Set δ = (δ 2,..., δ k, and let d = (d Λj : [l] j, 2 j l be a sequence satsfyng, for each 2 j k, d Λj d j for all [l] j. Let H = {H (j } k j=1 be an (l, k-complex, where H (1 = V 1 V l has n 0 n V n + 1 for each 1 l. If H has DEV(d, δ, then H (k H has l (H (k k = (1 ± µ many clques (k l., [l] j d Λj n l We now present a verson of Theorem 5.1 whch allows us to count copes of the (l, k- octohedron O (k = (k (U 1,..., U l, U 1 = = U l = 2, wthn an (l, k-complex H. Theorem 5.2 (Octohedral Countng Lemma. Let ntegers l k 2 be fxed. For all µ, d k > 0, there exsts δ k > 0 so that for all d > 0, there exsts δ > 0 so that,..., for all d 2 > 0, there exsts δ 2 > 0 and postve nteger n 0 so that the followng holds.

16 AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA 15 Set δ = (δ 2,..., δ k, and let d = (d Λj : [l] j, 2 j k be a sequence satsfyng that for all 2 j k and [l] j, d Λj d j. Let H = {H (j } k j=1 be an (l, k-complex, where H (1 = V 1 V k has n 0 n V n + 1, 1 l. If H has DEV(d, δ, then H (k H has EMB(O (k, H (k k = (1 ± µ [l] j d Λj n 2l many labeled partte-somorphc copes of the (l, k-octohedron O (k = (k (U 1,..., U l. We next present a type of extenson lemma (cf. Lemma 5.4, whch we wll descrbe n terms of the followng auxlary graph Γ. Defnton 5.3. For ntegers l k 2, let H = {H (j } k j=1 be an (l, k-complex, and let O(k be the (l, k-octohedron. We defne the octohedral-ncdence graph Γ = Γ k,l (H of H as follows. Set V ( Γ = l (H (k. For L, L V (Γ, put {L, L } Γ f, and only f, there exsts a labeled partte-embeddng ψ of O (k n H wth m ψ = L L,.e., L L nduces a copy of O (k n H (k. We now state the Octohedral Extenson Lemma. Theorem 5.4 (Octohedral Extenson Lemma. Fx ntegers l k 2. For all ζ, d k > 0, there exsts δ k > 0 so that for all d > 0, there exsts δ > 0 so that,..., for all d 2 > 0, there exst δ 2 > 0 and postve nteger n 0 so that the followng holds. Set δ = (δ 2,..., δ k, and let d = (d Λj : [l] j, 2 j k be a sequence satsfyng that, for all 2 j k and for all [l] j, d Λj d j. Let H = {H (j } k j=1 be an (l, k-complex, where H (1 = V 1 V l has n 0 n V n + 1 for each [l]. If H has DEV(d, δ and f Γ = Γ k,l (H s the octohedral-ncdence graph of H (cf. Defnton 5.3, then (1 all but ζ l (H (k clques L l (H (k satsfy k deg Γ (L = (1 ± ζ d 2j 1 [l] j n l ; (2 all but ζ l (H (k 2 pars of clques L L l (H (k satsfy k deg Γ (L, L = (1 ± ζ n l. d 2 2j 3 [l] j 6. The Negatve-Extenson Lemma In the prevous secton, we stated Countng and Extenson Lemmas correspondng to when a complex H has the devaton property DEV. In ths secton, we explore what happens when the property of devaton fals to hold. We gve our man result as Theorem 6.2, whch we call the Negatve-Extenson Lemma. We frst motvate ths result. Suppose H (k s a (k, k-cylnder wth underlyng (k, k 1-cylnder H (, where d = d(h (k H ( > 0. For δ > 0, suppose that (H (k, H ( does not have DEV(d, δ. Statement (2 of Fact 2.10 then guarantees that (H (k, H ( does not have COUNT emb (d, δ. As such, by Defnton 2.9 (recall (1 and (2, there exsts some S (k O (k = (k (U 1,..., U k so that EMB(S (k, (H (k, H ( (1 ± δd S(k EMB(O (, H (. (20

17 16 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT The Negatve-Extenson Lemma (Theorem 6.2 wll conclude that, as a result of (20, there are many k-tuples k (H ( whch belong to some unusual number of labeled partte embeddngs of S (k n H (k. To make our plan precse, we need some supportng concepts Supportng concepts, and the Negatve-Extenson Lemma. We use the followng notaton. For a (k, k-complex H = {H (j } k j=1, and for an nteger 1 k, let H( def = {H (j } j=1. Note that H( s a (k, -complex. Now, let Γ = Γ,k (H ( (21 be the octohedral-ncdence graph (cf. Defnton 5.3 of H (. Clearly, Γ k Γ Γ 2. (22 We also use the followng varant of the octohedral-ncdence graph Γ k, whch accomodates arbtary subhypergraphs S (k O (k = (k (U 1,..., U k. Defnton 6.1 (ncdence dgraph, anchor. Fx a (2, k, k-cylnder S (k O (k = (k (U 1,..., U k. Fx a k-tuple A = {a 1,..., a k }, where for each [k], a U. Let H (k = {H (j } k j=1 be a (k, k-complex. We defne the (S(k, A-ncdence dgraph Γ A (S (k = ΓA (S (k, H (k of H (k as follows. Set V ( Γ A (S (k = k (H (. For, V ( Γ A (S (k, put (, Γ A (S (k f, and only f, there exsts a labeled partte-embeddng ψ of S (k n (H (k, H ( (cf. Defnton 2.8 so that ψ(a = and m ψ =. We wll say that A s the anchor of Γ A (S (k, and we wll wrte Ā = (U 1 U k \ A. When workng wth the (S (k, A-ncdence dgraph Γ A (S (k = Γ A (S (k, H of a (k, k- complex H, we use the followng standard notaton. For, V ( Γ, we wrte N ΓA (S (k { ( = V ( Γ A (S (k : (, } Γ A (S (k, N ΓA (S (k (, = N ΓA (S (k ( N ΓA (S (k (, deg ΓA (S (k ( = N ΓA (S (k ( and deg ΓA (S (k (, = N ΓA (S (k (,. (23 Note that all neghborhoods and degrees defned above are out-neghborhoods and out-degrees. We now consder the followng statement EXT, whch consders a hypergraph S (k O (k, an anchor A for whch Ā S(k (cf. Defnton 6.1, and a (k, k-complex H (k. EXT A (S (k = EXT A (S (k, ξ, H (k. Fx S (k O (k = (k (U 1,..., U k, and fx an anchor A for whch Ā S(k (cf. Defnton 6.1. Let ξ > 0 be gven, and let H (k = {H (j } k j=1 be a (k, k-complex wth d [k] = d(h (k H ( > 0. Then, the followng condton holds: (1 If A S (k, then all but ξ H (k edges H H (k satsfy the followng mplcaton: If deg ΓA (S (k \{Ā}(H > ξ deg Γ (H, then deg ΓA (S (k (H = (1 ± ξd [k] deg ΓA (S(k\{Ā}(H;

18 AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA 17 (2 If A S (k, then all but ξ k (H ( clques k (H ( satsfy the followng mplcaton: If deg ΓA (S (k \{Ā}( > ξ deg Γ (, then deg ΓA (S (k ( = (1 ± ξd [k] deg ΓA (S(k\{Ā}(. For future purposes, t wll be convenent to have a compact presentaton of the statement EXT A (S (k = EXT A (S (k, ξ, H (k (see (26 below. To that end, let { A (k = A (k (S (k, A, H (k H = (k f A S (k, k (H ( f A S (k (24. In the language of A (k, we wll combne Condtons (1 and (2 of EXT A (S (k nto one presentaton, as follows. Set { A (k bad = A(k bad (S(k, A, ξ, H (k = A (k : deg ΓA (S (k\ā( > ξ deg Γ ( } but deg ΓA (S (k ( (1 ± ξd [k] deg ΓA (S (k\ā(. (25 Then, EXT A (S (k, ξ, H (k s true A (k bad < ξ A(k. (26 We now state the man result of the secton, the Negatve-Extenson Lemma. Theorem 6.2 (The Negatve-Extenson Lemma. Let nteger k 2 be fxed. For all d k, δ k > 0, there exsts ξ > 0 so that for all d > 0, there exsts δ > 0 so that,..., for all d 2 > 0, there exst δ 2 > 0 and postve nteger n 0 so that the followng holds. Set δ = (δ 2,..., δ. Let H = H (k be a (k, k-complex wth densty sequence d k, as gven n Setup 2.15, where n n 0. Suppose d k satsfes that, for each 2 j k, d Λj d j for all [k] j. Assume that (1 H ( has DEV(d, δ, but that (2 (H (k, H ( does not have DEV(d [k], δ k. Then, there exsts a hypergraph S (k O (k = (k (U 1,..., U k so that, whenever an anchor A satsfes Ā S(k, the statement EXT A (S (k, ξ, H (k s false. In other words, the hypergraphs A (k = (S (k, A, H (k and A (k bad = A(k bad (S(k, A, ξ, H (k satsfy A (k bad ξ A(k. We proceed to defne the constants for Theorem The constants of Theorem 6.2. Let k 2 be a fxed nteger, and let d k, δ k > 0 be gven. We defne the constant ξ promsed by Theorem 6.2 by 1 ξ = 100k2 k δ kd 2k k. (27 Let d > 0 be gven. We formally defne the constant δ n upcomng (29, but we frst motvate how we choose t. To that end, defne auxlary constants (cf. (27 µ = 1/2 and ζ = ξd k2. (28 Recall from the hypothess of Theorem 6.2 that we wll be workng wth a (k, k 1-complex H ( = {H (j } whch has DEV(d, δ, where the constants d k 2,..., d 2 of d and

19 18 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT the constants δ,..., δ 2 of δ wll be dsclosed below. For such a complex H (, we want δ > 0 to be small enough so that the followng condtons are satsfed (cf. (28: (a we can estmate k (H ( wthn an error of 1 ± µ; (b we can estmate EMB(O (, H ( wthn an error of 1 ± µ; (c all but ζ k (H ( clques k (H ( satsfy deg Γ ( = (1 ± ζ d 2j 1 [k] j n k. To guarantee that (a, (b, and (c above are satsfed, we need δ > 0 to be small enough to enable applcatons of Theorems 5.1, 5.2, and 5.4. Wth d gven above, and wth µ = 1/2 from (28, let δ Thm.5.1, k 1 = δ Thm.5.1 (l = k, k 1, µ = 1/2, d > 0 and δ Thm.5.2, k 1 = δ Thm.5.2 (l = k, k 1, µ = 1/2, d > 0 be the constants guaranteed by Theorems 5.1 and 5.2. Wth d gven above, and wth ζ from (28, let δ Thm.5.4, k 1 = δ Thm.5.4 (l = k, k 1, ζ = ζ, d > 0 be the constant guaranteed by Theorem 5.4. Now, set δ = mn {δ Thm.5.1, k 1, δ Thm.5.2, k 1, δ Thm.5.4, k 1 } (29 whch concludes our defnton of the promsed constant δ. Inductvely, assume d, δ,..., d, δ, d 1 have been dsclosed, for a fxed nteger satsfyng 3 k 1. Moreover, assume that we have defned auxlary constants (cf. (28 ζ = ξd k2, ζ k 2 = ξd ( k 2 d ( k k 22 k 2 k 2,... ζ 1 = ξ j= 1 d (k j2 j j. (30 We defne δ 1 smlarly to how we defned δ (cf. (29. In partcular, we want δ 1 > 0 to be small enough so that (a and (b above are satsfed wth µ = 1/2. These tasks are handled by Theorems 5.1 and 5.2, whch have the followng common quantfcaton of constants: µ, d, δ :... d 1, δ 1 :... Wth µ = 1/2 from (28, and wth d, δ,..., d 1 nductvely dsclosed above, let δ Thm.5.1, 1 = δ Thm.5.2 (l = k, k 1, µ = 1/2, d, δ,..., d, δ, d 1 > 0 and δ Thm.5.2, 1 = δ Thm.5.2 (l = k, k 1, µ = 1/2, d, δ,..., d, δ, d 1 > 0 be the constants guaranteed by Theorems 5.1 and 5.2. We also want δ 1 > 0 to be small enough so that (c above s satsfed wth ζ from (28. Moreover, we want δ 1 > 0 to be small enough so that the followng sequence (c of condtons s satsfed (cf. (30: (c all but ζ k (H ( clques k (H ( satsfy deg Γ ( = (1 ± ζ d 2j 1 [k] j n k ;

20 AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA 19 all but ζ k 2 k (H (k 2 clques k (H (k 2 satsfy k 2 deg Γk 2 ( = (1 ± ζ k 2. d 2j 1 [k] j n k ; all but ζ 1 k (H ( 1 clques k (H ( 1 satsfy 1 deg Γ 1 ( = (1 ± ζ 1 d 2j 1 [k] j n k. To guarantee that the sequence (c of condtons above wll be satsfed, we fx an nteger h satsfyng 1 h, and we appeal to Theorem 5.4, whch has the followng quantfcaton of constants: ζ, d h, δ h : d h 1, δ h 1 :... d 1, δ 1 :... Wth d h, δ h,..., d 1, δ 1 nductvely dsclosed above, and wth ζ = ζ h from (30, let δ Thm.5.4, 1, h = δ Thm.5.4 (l = k, h, ζ = ζ h, d h, δ h,..., d, δ, d 1 be the constant guaranteed by Theorem 5.4. Set Fnally, set δ Thm.5.4, 1 = mn {δ Thm.5.4, 1, h : 1 h k 1}. δ 1 = mn {δ Thm.5.1, 1, δ Thm.5.2, 1, δ Thm.5.4, 1 }. (31 We contnue ths way untl δ 2 s reached. Ths concludes our defntons of the constants The argument for Theorem 6.2. Set δ = (δ 2,..., δ, where each δ j, 2 j k 1, was defned n (31. Let H (k be a (k, k-complex wth densty sequence d k, as gven n Setup 2.15, where n n 0. Suppose d k satsfes that, for each 2 j k, d Λj d j for all [k] j, where d j was gven above. Suppose that H ( has DEV(d, δ, but that (H (k, H ( does not have DEV(d [k], δ k. Theorem 6.2 promses a hypergraph S (k O (k = (k (U 1,..., U k so that, for any anchor A for whch Ā S(k (cf. Defnton 6.1, the statement EXT A (S (k, ξ, H (k s false. We begn our argument by defnng the promsed hypergraph S (k Defnng the hypergraph S (k. Frst, we appeal to (20, and take any hypergraph S (k O (k for whch EMB(S (k, (H (k, H ( (1 ± δk d S(k EMB(O (k, H (. (32 Indeed, Assumpton (2 of our hypothess says that (H (k, H ( does not have DEV(d [k], δ k. As such, Statement (2 of Fact 2.10 gves that (H (k, H ( does not have COUNT emb (d [k], δ k. Thus, some S (k O (k satsfyng (32 s guaranteed to exst by Defnton 2.9. Second, take S (k EMB(S (k mn, (H(k, H ( [k] mn S(k to be an edge-mnmal subhypergraph for whch ( δ k 1 ± d S(k mn EMB(O (, H (. (33 2 S(k S (k mn [k]

21 20 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT (Note that n (33, we requre the error δ k /2 S(k S (k mn to decrease as S (k mn decreases. Note that S (k mn must exst, because S(k tself satsfes (32. Note also that S (k mn, because EMB(, (H (k, H ( = (1 ± 0d 0 EMB(O ( [k], H (. Snce S (k mn s edge-mnmal w.r.t. (33, we have that, for each e S(k mn, ( δ k 1 ± d S(k mn 1 EMB(O (k, H (. (34 EMB(S (k mn \ {e}, (H(k, H ( = For smplcty of notaton, we shall wrte Then, we may rewrte (33 as δ k := δ k 2 S(k S (k mn +1 [k] and S (k := S (k 2 S(k S (k mn mn. (35 EMB(S (k, (H (k, H ( (1 ± δ k EMB(O ( d S(k, H (, (36 and we may rewrte (34 as, for each e S (k, EMB(S (k \ {e}, (H (k, H ( = ( 1 ± δ k 2 [k] d S(k 1 [k] EMB(O (, H (. (37 Ths concludes our defnton of the promsed hypergraph S (k. We pause to say a word about the nequalty n (36. We have that ether EMB(S (k, (H (k, H ( < (1 δ k d S(k [k] EMB(O (, H (, or EMB(S (k, (H (k, H ( > (1 + δ k EMB(O ( d S(k, H (. In our proof, t wll be symmetrc to handle ether stuaton above. We therefore assume, w.l.o.g., that the latter holds: EMB(S (k, (H (k, H ( > (1 + δ k EMB(O ( d S(k, H (. (38 We proceed to develop a proof by contradcton. Assume the hypergraph S (k from (38 doesn t have the desred property of Theorem 6.2. In partcular, assume that there exsts an anchor A, where Ā S(k, for whch the statement EXT A (S (k, ξ, H (k s true. Wth ths assumpton, we wll prove the followng. Clam 6.3. Assumng the statement EXT A (S (k, ξ, H (k s true for some Ā S(k, we have EMB(S (k, (H (k, H ( ( δ k d S(k EMB(O (, H (. Now, the bound n Clam 6.3 s a drect contradcton wth the bound n (38. Thus, t must be the case that for any anchor A, where Ā S(k, the statement EXT A (S (k, ξ, H (k s false, as promsed by Theorem 6.2. Thus, to complete the proof of Theorem 6.2, t only remans to prove Clam 6.3. [k] [k] [k]

22 AN ALGORITHMIC HYPERGRAPH REGULARITY LEMMA Proof of Clam 6.3. Assume that the statement EXT A (S (k, ξ, H (k s true for some anchor A wth Ā S(k. Recall that n (24 (26, we abbrevated the truth of the statement EXT A (S (k, ξ, H (k n terms of the followng hypergraphs A (k and A (k bad : { A (k = A (k (S (k, A, H (k H = (k f A S (k, k (H ( f A S (k, { A (k bad = A(k bad (S(k, A, ξ, H (k = A (k : deg ΓA (S (k\ā( > ξ deg Γ ( } but deg ΓA (S (k ( (1 ± ξd [k] deg ΓA (S (k\ā(. Recall from (26 that our assumpton that EXT A (S (k, ξ, H (k s true s equvalent to A (k bad < ξ A(k ξ k (H (. (39 Defne also the sets { A (k good = A(k good (S(k, A, ξ, H (k = A (k : deg ΓA (S (k\ā( > ξ deg Γ ( } and deg ΓA (S (k ( = (1 ± ξd [k] deg ΓA (S (k\ā(, (40 and { } A (k 0 = A (k 0 (S(k, A, ξ, H (k = A (k : deg ΓA (S (k\ā( < ξ deg Γ (. (41 Note that A (k = A (k good A(k bad A(k 0 (42 s a partton. Usng the partton A (k = A (k good A(k bad A(k 0 from (42, observe that (recall Defnton 2.8 EMB(S (k, (H (k, H ( = (42 = A (k good A (k deg ΓA (S (k ( deg ΓA (S (k ( + A (k bad We now bound each of the sums above. Frst, usng the defnton of A (k good n (40, we have deg ΓA (S (k ( (1 + ξd [k] A (k good = (1+ξd [k] EMB(S (k \{Ā}, (H(k, H ( (37 ( 1 + 2ξ + δ k 2 deg ΓA (S (k ( + deg ΓA (S(k\{Ā}( A (k d S(k [k] (27, (35 A (k 0 ( (1+ξ 1 + δ k 2 [k] EMB(O (, H ( ( 1 + 2δ k 3 d S(k [k] d S(k deg ΓA (S (k (. (43 EMB(O (, H ( EMB(O (, H (. (44

23 22 BRENDAN NAGLE, VOJTĚCH RÖDL, AND MATHIAS SCHACHT (To see the last nequalty, (27 gves ξ < δ k /(2 k 6, and (35 gves δ k /2 k < δ k. Second, we take A (k 0 deg ΓA (S (k ( A (k 0 deg ΓA (S(k\{Ā}(, snce every labeled partte-embeddng of S (k n H (k s also a labeled partte-embeddng of S (k \ {Ā} n H(k. Usng the defnton of A (k 0 n (41, we have A (k 0 Thrd, we take deg ΓA (S (k ( A (k 0 A (k bad deg ΓA (S(k\{Ā}( ξ deg Γ ( = ξemb(o (, H (. (45 A (k deg ΓA (S (k ( A (k bad deg Γ (, (46 snce every labeled partte-embeddng of S (k n H (k s also a labeled partte-embeddng of O ( n H (. More strongly, we have the followng bound (whch we prove n a moment. Fact 6.4. A (k bad deg Γ ( 8(k 1ξ EMB(O (, H (. Applyng the bounds of (44 (46 and the bound of Fact 6.4 to (43, we nfer EMB(S (k, (H (k, H ( (( δ k d S(k [k] + ξ + 8(k 1ξ EMB(O (, H ( ( δ k + 8kξd 2k k d S(k [k] EMB(O (, H (, (47 where we used S (k 2 k and d [k] d k from the hypothess of Theorem 6.2. Now, snce we have 8kξd 2k k (27 < k δ (35 k < 1 12 δ k, EMB(S (k, (H (k, H ( ( < δ k d S(k [k] EMB(O (, H (, as promsed by Clam 6.3. Thus, t only remans to prove Fact Proof of Fact 6.4. We frst outlne the man dea of how we bound A (k deg Γ (. bad To begn, we dvde the k-tuples A (k bad nto two classes: those for whch deg Γ ( s not too large, and those for whch t s. More generally, we frst partton the set of k-tuples